How to find the solution to this system of integer variables?

Suppose that some $x_j <3$. Wlog we can assume $j=1$. Then $$ \sum_{i=1}^{n} x_i = x_1 + \sum_{i=2}^{n}x_i \le 2 + 3(n-1) < 3n. $$ So we have a contradiction.

$$\sum_{i=1}^nx_i \leq \sum_{i=1}^n3=3n$$ and the equality holds only when $x_i=x_j$ for all $i,j$.

So the only solution is $x_i=3$ for all $i$

Suppose at least $1 \ x_i$ is less than $3$. Then there must be at least one $x_j$ strictly greater than $3$ to make up for $3n$ . Hence a contradiction